Representability of the local motivic Brouwer degree

TL;DR

This paper investigates which quadratic forms can be realized as local degrees of maps with isolated zeros, revealing limitations over certain fields and classifying forms of low rank, with implications for algebraic topology and quadratic form theory.

Contribution

It provides a complete classification of quadratic forms of rank at most 7 that are representable as local degrees over fields of characteristic not 2, and demonstrates that not all forms are representable.

Findings

Not all quadratic forms are representable as local degrees over some fields.

Local degrees of low rank predominantly have hyperbolic summands.

Complete classification for forms of rank ≤ 7 over fields with characteristic ≠ 2.

Abstract

We study which quadratic forms are representable as the local degree of a map $f : A^n \to A^n$ with an isolated zero at $0$, following the work of Kass and Wickelgren who established the connection to the quadratic form of Eisenbud, Khimshiashvili, and Levine. Our main observation is that over some base fields $k$, not all quadratic forms are representable as a local degree. Empirically the local degree of a map $f : A^n \to A^n$ has many hyperbolic summands, and we prove that in fact this is the case for local degrees of low rank. We establish a complete classification of the quadratic forms of rank at most $7$ that are representable as the local degree of a map over all base fields of characteristic different from $2$. The number of hyperbolic summands was also studied by Eisenbud and Levine, where they establish general bounds on the number of hyperbolic forms that must appear in a quadratic form that is representable as a local degree. Our proof method is elementary and constructive in the case of rank 5 local degrees, while the work of Eisenbud and Levine is more general. We provide further families of examples that verify that the bounds of Eisenbud and Levine are tight in several cases.